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Distribution Free vs. Non-distribution Free Methods in Factor

Analysis

Nicola Ritter, M.Ed.EPSY 643: Multivariate Methods

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.

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Top 5 Take Away Points

1. Extracted factors from a covariance matrix are a function of correlations and standard deviations.

2. Different factors may be extracted based on the matrix of associations selected.

3. Correlational statistics represented in matrices address different questions.

4. Factors are sensitive to the information available in a given correlation statistic.

5. Factors are extracted from a matrix of associations.

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5. Factors are extracted from a matrix of associations.

• Scores on measured variables are used to compute matrices of bivariate associations.

• i.e. Covariance matrix or correlation matrixEven given only a matrix of associations, all steps

in factor analysis can be completed (except for calculating the factor scores).

  VAR1 VAR2 VAR3 VAR4 VAR5 VAR6

VAR1 1          

VAR2   1        

VAR3     1      

VAR4       1    

VAR5         1  

VAR6           1

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What are the different types of correlation statistics?

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4. Factors are sensitive to the information available in a given correlation statistic.

continuous rpb r

rank ρ

categorical Ф rpb

nominal ordinal interval

Bivariate Correlation Coefficients

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Pearson r Correlation Matrix

• Most commonly used in EFA• Default in most statistical packages

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Pearson’s r vs. Spearman’s rho

Pearson’s r• Variables are intervally

scaled

Spearman’s ρ• Variables are at least

ordinally scaled.

If the data are intervally scaled, either correlation coefficient could be used.

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Which correlation coefficient do we use?

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Pearson r AssumptionParticipant X x̄� x Y Ybar y xy

1 3 4.0 -1.0 3 33.0 -30.0 30.0

2 4 4.0 0.0 4 33.0 -29.0 0.0

3 5 4.0 1.0 92 33.0 59.0 59.0

Sum 12.00 99.00 89.0

Mean 4.00 33.00

SD 1.00 51.10

COV 44.50

r 0.87

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Spearman rho Assumption

Participant X x̄� x Y Ybar y xy

1 1 2.0 -1.0 1 2.0 -1.0 1.0

2 2 2.0 0.0 2 2.0 0.0 0.0

3 3 2.0 1.0 3 2.0 1.0 1.0

Sum 6.00 6.00 2.0

Mean 2.00 2.00

SD 1.00 1.00

COV 2.00

r 1.00

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3. Correlational statistics represented in matrices address different questions.

Spearman’s rho1. Addresses the question,

“How well do the two variables order the cases in exactly the same (or the opposite) order?” (Thompson, 2004, p. 130)

Pearson r1. Addresses the same

question AND2. Addresses the question,

“To what extent do the two variables have the same shape?” (Thompson, 2006, p. 130)

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2. Different factors may be extracted based on the matrix of associations selected.

I. Pearson r Correlation Matrix II. Spearman’s rho Correlation MatrixIII. Covariance Matrix

Data from Thompson, 2004, Appendix A, ID 001-007 & PER1-PER6

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Pearson’s r Matrix

  PER1 PER2 PER3 PER4 PER5

PER1 1.000 0.716 0.654 0.418 0.205

PER2 0.716 1.000 0.720 0.345 0.226

PER3 0.654 0.720 1.000 0.764 0.812

PER4 0.418 0.345 0.764 1.000 0.858

PER5 0.205 0.226 0.812 0.858 1.000

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Syntax with Factor Analysis with Pearson’s r Matrix

FACTOR /VARIABLES PER1 PER2 PER3 PER4 PER5 /MISSING LISTWISE /ANALYSIS PER1 PER2 PER3 PER4 PER5 /PRINT INITIAL EXTRACTION ROTATION /CRITERIA MINEIGEN(1) ITERATE(25) /EXTRACTION pc /CRITERIA ITERATE(25) /ROTATION varimax /METHOD=CORRELATION.

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Output of Factor Analysis with Pearson’s r Matrix

Notes. Principal components extraction, varimax-rotated factor matrix

Factor 1 Factor 2 h²

0.169 0.902 0.841

0.161 0.920 0.872

0.749 0.627 0.955

0.911 0.242 0.888

0.985 0.064 0.974

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Spearman’s rho Matrix

  PER1 PER2 PER3 PER4 PER5

PER1 1.000 0.687 0.647 0.392 0.210

PER2 0.687 1.000 0.732 0.283 0.216

PER3 0.647 0.732 1.000 0.574 0.761

PER4 0.392 0.283 0.574 1.000 0.781

PER5 0.210 0.216 0.761 0.781 1.000

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Syntax with Factor Analysis Spearman’s rho Matrix

NONPAR CORR /VARIABLES=PER1 PER2 PER3 PER4 PER5 /PRINT=SPEARMAN /MATRIX=OUT(*) /MISSING=LISTWISE .

RECODE rowtype_ ('RHO'='CORR') .EXECUTE .

FACTOR /MATRIX=IN(cor=*) /ANALYSIS PER1 PER2 PER3 PER4 PER5 /PRINT INITAL EXTRACTION ROTATION /CRITERIA MINEIGEN(1) ITERATE(25) /EXTRACTION pc /CRITERIA ITERATE(25) /ROTATION varimax /METHOD=CORRELATION .

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Output of Factor Analysis with Spearman’s rho Matrix

Notes. Principal components extraction, varimax-rotated factor matrix

Factor 1 Factor 2 h²

0.882 0.159 0.804

0.924 0.117 0.868

0.697 0.644 0.900

0.199 0.881 0.815

0.107 0.969 0.951

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2. Different factors may be extracted based on the matrix of associations selected.

Pearson’s r Spearman’s rho

Notes. Principal components extraction, varimax-rotated factor matrix

Table 1 Table 2

Factor 1 Factor 2 h²

0.169 0.902 0.841

0.161 0.920 0.872

0.749 0.627 0.955

0.911 0.242 0.888

0.985 0.064 0.974

Factor 1 Factor 2 h²

0.882 0.159 0.804

0.924 0.117 0.868

0.697 0.644 0.900

0.199 0.881 0.815

0.107 0.969 0.951

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1.Extracted factors from a covariance matrix are a function of correlations and standard deviations.

• Matrix most commonly used in CFA• Covariance is Pearson r with standard deviations

removed

rXY = COVXY / (SDX * SDY)COVXY = rXY * SDX * SDY

• Jointly influenced by:1. Correlation between the two variables2. Variability of the first variable3. Variability of the second variable

Thompson (2004)

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Syntax with Factor Analysis Covariance Matrix

FACTOR /VARIABLES PER1 PER2 PER3 PER4 PER5 /MISSING LISTWISE /ANALYSIS PER1 PER2 PER3 PER4 PER5 /PRINT INITIAL EXTRACTION ROTATION /CRITERIA MINEIGEN(1) ITERATE(25) /EXTRACTION pc /CRITERIA ITERATE(25) /ROTATION varimax /METHOD=cov.

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Covariance Matrix

  PER1 PER2 PER3 PER4 PER5

PER1 1.905 1.238 1.024 0.667 0.381

PER2 1.238 1.571 1.024 0.500 0.381

PER3 1.024 1.024 1.286 1.000 1.238

PER4 0.667 0.500 1.000 1.333 1.330

PER5 0.381 0.381 1.238 1.333 1.810

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Output of Factor Analysis with Covariance Matrix

Factor 1 Factor 2 h²

0.163 0.923 0.878

0.174 0.898 0.836

0.760 0.615 0.955

0.900 0.250 0.873

0.990 0.055 0.982

Notes. Principal components extraction, varimax-rotated factor matrix

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Matrices Sensitivity to Different Aspects of the Data

FA with Pearson r Matrix

FA with Spearman rho Matrix

FA with Covariance Matrix

Factor 1

Factor 2 h²

Factor 1

Factor 2 h²

Factor 1

Factor 2 h² 

0.169 0.902 0.841 0.882 0.159 0.804 0.163 0.923 0.878

0.161 0.920 0.872 0.924 0.117 0.868 0.174 0.898 0.836

0.749 0.627 0.955 0.697 0.644 0.900 0.760 0.615 0.955

0.911 0.242 0.888 0.199 0.881 0.815 0.900 0.250 0.873

0.985 0.064 0.974 0.107 0.969 0.951 0.990 0.055 0.982

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Top 5 Take Away Points

1. Extracted factors from a covariance matrix are a function of correlations and standard deviations.

2. Different factors may be extracted based on the matrix of associations selected.

3. Correlational statistics represented in matrices address different questions.

4. Factors are sensitive to the information available in a given correlation statistic.

5. Factors are extracted from a matrix of associations.

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References

Gorsuch, R.L. (1983). Factor analysis (2nd ed.). Hillsdale, NJ: Erlbaum.

Thompson, B. (2004). Exploratory and confirmatory factor analysis: Understanding concepts and applications. Washington, DC: American Psychological Association.

Thompson, B. (2006). Foundations of behavioral statistics: An insight-based approach. New York, NY: Guilford.